Questions tagged [calculus-of-variations]
This tag is for problems relating to the calculus of variations that deal with maximizing or minimizing functionals. This problem is a generalization of the problem of finding extrema of functions of several variables. In fact, these variables will themselves be functions and we will be finding extrema of “functions of functions” or functionals.
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deriving the variation (small change) of the extension of a string [closed]
what is the process to get from the first line (extension of a string) to the second line (small deviation of the extension in the string). Thanks
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Minimizing $(f')^2 + f^2$
I've confused myself when thinking about the following variational problem:
\begin{equation}
\min_{f} \int_0^T \left([f'(x)]^2 + [f(x)]^2\right)\,dx \qquad f(0) = 1, f'(0) = 0. \tag{*}
\end{equation}
...
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Variational Problem.
How to solve variational problem $$I(y)=\int_0^1 [(y’)^2-y|y|y’+xy]dx,y(0)=y(1)=0?$$
I tried by Euler equation, which is $$ -2|y|y’+x-\frac{d}{dx}(2y’-y|y|)=0$$ Now stuck. Unable to creat ...
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How should I use the Jacobi equation to determine the nature of this stationary path? [closed]
Let $n>1$ be a positive integer such that the functional $S[y]=\int_{0}^{1}(y')^{n}e^{y}dx, y(0)=1, y(1)=A>1$ has a stationary path given by $y=n\cdot ln(cx+e^{1/n})$, where $c=e^{A/n}-e^{1/n}$. ...
- 177
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Prove the lower semi-continuity of inner products [duplicate]
Let $\mathcal{H}$ be a Hilbert space equipped with inner product $(\cdot,\cdot)$. Suppose that a sequence $u_n\rightarrow u$ weakly in $(\mathcal{H},(\cdot,\cdot))$, I want to prove that $$(u,u)\le \...
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Broken Extremal.
Consider the Functional $$J(y)=\int_0^2(1-y’^2)^2dx$$ defined on $$\{y\in C[0,2]\mid \text{y is piecewise $C^1$ and $y(0)=y(2)=0$}\}$$ Let $y_e$ be a minimizer of the above functional. Then $y_e$ has
$...
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2
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How should I find the value of $a$ such that the functional can be written as the sum of two functionals?
Let the two coupled Euler-Lagrange equations giving the stationary path of the functional $S[y_{1}, y_{2}]=\int [y_{1}'^2+2y_{2}'^2+(2y_{1}+y_{2})^2]dx$ be $y_{1}''-2(2y_{1}+y_{2})=0$ and $2y_{2}''-(...
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Riesz representation theorem in the derivative of functional.
Definition 1 (Functional derivative): Given a function $f \in \mathcal{F}$, the functional derivative of $F$ at $f$, denoted $\frac{\partial F}{\partial f}$, is defined to be the function for which:
$$...
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Relation of $n$-th variation, Gâteaux and Fréchet derivatives
Let $E$ be a Banach space, $U \subset E$ be an open subset of $E$ and $f: U \to \mathbb{R}$ a given function. For fixed $x_{0}, h \in E$, there exists $t_{0} \ge 0$ such that $x_{0} + (-t_{0},t_{0})h \...
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How to show that this functional has this stationary path by solving this second-order differential equation?
Let $n>1$ be a positive integer. Show that the functional $$S[y]=\int_{0}^{1}(y')^{n}e^{y}dx, \quad y(0)=1,\quad y(1)=A>1,$$ has a stationary path given by $$y=n \ln(cx+e^{1/n}),$$ where $$c=e^{...
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Computing limit of an energy of a probability measure
For $c\in (0,1)$, consider the probability density
$$f_c := \frac{1-c^2}{1+c^2-2c\cos(2\pi x)}.$$
Goal. Evaluate the limit as $c\rightarrow 1$ of
$$E(f_c) := \int_{-\frac12}^{\frac12}\log(f_c)f_c dx -...
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A question about the derivative of functional.
I'm reading Existence of solutions to a higher dimensional mean-field equation on manifolds, in this paper they defined
$$
E:=\left\{u \in H^m(M): \int_M u d \mu_g=0\right\},
$$
and
$$\|u\|:=\left(\...
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How do I continue with this differential equation from the Brachistochrone problem with friction to get the correct parametric solution?
I'm stuck in getting the parametric solution to the Brachistochrone problem WITH friction. I've followed up to step 30, but I don't understand how the results in 32 and 33 were calculated.
In the ...
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Is there a potential energy formula for a catenary curve?
I am trying to find the minimum energy point of a catenary curve. Can anyone please let me know whether there is a potential energy or energy formula for a catenary curve ?
Especially I would like to ...
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Find stationary point of non-canonical form in calculus of variation
Let $x \in [0, 1]^{10}$ be a ten dimensional vector. I want to maximize the functional with the form:
$$G(f) = \int_{0 \leq y_1 \leq x_1 \leq 1, x_i, y_i \in [0, 1], i \neq 1} f(x) f(y) (1 + 3 x_1^2 - ...
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2
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How should I find this first-integral of a functional?
Let $0<a<b$. Consider the functional $$S[y]=\int_{a}^{b}x^5(y'^2-\frac{2}{3}y^3)dx$$ Prove that a first-integral of $S[y]$ is $$4x^5yy'+x^6(y'^2+\frac{2}{3}y^3)=c,$$ where $c$ is constant, ...
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How should I show that $S[y]$ is invariant under the scale transformation?
Let $0<a<b$. Consider the functional $S[y]=\int_{a}^{b}x^5(y'^2-\frac{2}{3}y^3)dx$. Show that $S[y]$ is invariant under the scale transformation $\bar{x}=\alpha x, \bar{y}=\beta y$, where $\...
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Finding approximation to eigenvalues from minizing Rayleigh quotient and from diagonalization
Suppose I want to find an approximation to the smallest eigenvalue of $y''=\lambda y$, with $y(0)=y(1)=0$.
One way to do it is to write the ansatz $y=x(1-x)+ax^2(1-x)^2$, compute the Rayleigh quotient,...
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Dimension of null space of an operator $T$
Let, $K(x,y)$ be a kernal in $[0,1]\times [0,1]$ defined as $K(x,y)=\sin(2\pi x)\sin(2\pi y)$. Consider the integral operator
$$T(u)(x)=\int_0^1 u(y)K(x,y)\,dy$$
where, $u\in C[0,1]$. Which of the ...
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How should I find conditions on the constants $b_{0}, c_{0}, \omega, \gamma$?
According to an economic model, the budget $b(t)$ at time $t\geq 0$ in a household is chosen to maximise the lifetime utility $U[b]=\int_{0}^{\infty}e^{-\beta t}u(c(t))dt$, where $u(c)\geq 0$ is the ...
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How should I calculate the Euler-Lagrange equation in this problem?
According to an economic model, the budget $b(t)$ at time $t\geq 0$ in a household is chosen to maximise the lifetime utility $U[b]=\int_{0}^{\infty}e^{\beta t}u(c(t))dt$, where $u(c)\geq 0$ is the ...
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Minimizing $\int_1^2\frac{1}{x}\sqrt{1+u^{\prime}(x)^2}dx$
I want to minimize $\int_1^2\frac{1}{x}\sqrt{1+u^{\prime}(x)^2}dx$ so that $u(1)=0$ and $u(2)=1$.
Using the Euler-Lagrange-equation I obtain $u(x)=2-\sqrt{5-x^2}$.
Futhermore, the second variation of $...
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What variational problem does the parabolic suspension bridge solve?
The catenary curve $y(x)$ minimizes the gravitational potential energy
$$\int \rho g y ds=\int \rho g y \sqrt{1+y'^2}dx,$$
subject to a fixed length, $L=\int \sqrt{1+y'^2}dx.$
It is known that in a ...
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Derivative in Euler-Lagrange Equations
In van Brunt's "The Calculus of Variations" the author derives the Euler-Lagrange equation. To start, the author notes that the first variation $\delta J(\eta, y)$ of a functional $J: C^2[...
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A problem with the calculus of variations
$$\int\limits_{0}^{1}x(t)\sqrt{1+u^2(t)}dt\to inf, \; \dot{x}(t)=u(t), \; u\in PC^1[0,1], \; x(t)\geq 0, \; |u(t)|\leq k, \; x(\pm 1)=R$$
I'm having trouble finding an optimisation. I decided to use ...
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Finding two coupled Euler-Lagrange equations giving the stationary path of the functional $S[y_{1},y_{2}]=\int[y_{1}'^2+2y_{2}'^2+(2y_{1}+y_{2})^2]dx$
Find the two coupled Euler-Lagrange equations giving the stationary path of the functional
$$S[y_{1}, y_{2}]=\int \left[y_{1}'^2+2y_{2}'^2+(2y_{1}+y_{2})^2\right]dx$$
Here's my work:
Consider the ...
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Find stationary point of sum of the square of integral
Let $x \in [0, 1]^{10}$ be a ten dimensional vector. I want to maximize the functional with the form:
$$G(f) = (\int_{x} f(x) x_1 dx)^2 + (\int_{x} f(x) e^{x_2} dx)^2$$
with constraints
$$\int_{x} f(x)...
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Example of an optimization problem with time-dependent logical constraints?
I am wondering if there are any examples of an optimal control problem in which a design variable is constrained based on the instantaneous time. Like for example say $x(t)\forall t\in[t_0,t_f]$ is a ...
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Find the minimum of integral
For $g(x):\Bbb R \rightarrow [0,\infty],$ if $$\int_{-\infty}^{\infty} g(x)\, dx= \int_{-\infty}^{\infty} x^2g(x)\, dx=C$$
find the minimum value of $\int_{-\infty}^{\infty} \ln(g(x))g(x)\, dx$.
...
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The relation between the Euler-Lagrange equation and the Beltrami identity
This question is specifically about deriving the Beltrami identity.
Just to give this question context I provide an example of a problem that is solved with Calculus of Variations: find the shape of a ...
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Lower Semicontinuity of $L^p$ norms with varying exponents
In a previous post (see continuity of $L^p$ norms with respect to $p$) it is shown that in a measure space $(\Omega,\Sigma,\mu)$, if $1\leq p_0\leq p\leq p_1\leq+\infty$, then the function $\Phi\...
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I am trying to find the time for different paths for a traversing ball between two points under the influence of gravity? But, I am getting infinity?
I am trying to find the time for different paths (functions) for a traversing ball between two points under the influence of gravity, but I am often getting strange answers where the integrand tends ...
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Trying to find the time for different paths for a traversing ball between two points under the influence of gravity, but I need some help? [duplicate]
I am trying to find the time for different paths for a traversing ball between two points under the influence of gravity, but I am getting strange results for a reason I don't understand, I feel like ...
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1
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How to find the range of values of $A$ in this derivation? Please check my work here.
Consider the functional $S[y]=\int_{1}^{2}ln(1+x^2y')dx, y(1)=0, y(2)=A$, where $A$ is a constant and $y$ is a continuously differentiable function for $1\leq x\leq 2$. Let $h$ be a continuously ...
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Cotangent lift of the embedding is zero
I am reading the work on variatonal collision integrators from the programmers point of view, since I want to implement it is the software. The work is publicly available here: caltech. I want to ...
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Definition of functional differential
According to Wikipedia, there are two definitions for the differential of a functional (also called the variation). The first is the Fréchet derivative of the functional and the second the Gateaux ...
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Papers or lectures on finding the solution of elliptic PDE which is the saddle point of its energy functional.
Can you recommend me some papers or lectures on finding the solution of elliptic PDE which is the saddle point of its energy functional.
I glance over some methods including mountain pass theorem and ...
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Minimizing a functional subject to boundary conditions
I want to find a smooth function $y : [0,1] \to \mathbb{R}$ that minimizes
$$
S = \int_0^1 (y(x)y'''(x) + 3 y'(x) y''(x))^2 dx
$$
subject to the constraints / boundary conditions: (i) $y(0)=1$, (ii) $...
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How should I prove this functional? What technique should I apply?
Consider the functional $S[y]=\int_{1}^{2}ln(1+x^2y')dx, y(1)=0, y(2)=A$, where $A$ is a constant and $y$ is a continuously differentiable function for $1\leq x\leq 2$. Let $h$ be a continuously ...
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0
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Finding conditions on a constant for coercivity using Lagrange Multipliers
I need to find conditions on $\alpha$ such that the bilinear, symmetric form $a(u,v) = \int \nabla u \nabla v + \alpha \int u v$ is coercive in order to use Lax-Milgram over the Sobolev space $H_{0}^{...
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How to prove that the only curve that minimises the distance between two points in Euclidean space is a straight line?
I am able to prove that one curve in $C^1([0,1], \mathbb{R})$, $r$, that has minimal length connecting two points is a straight line. If these points are $p, q \in \mathbb{R}^n$, then the distance is $...
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How should I derive the Euler-Lagrange equation by using the Gateaux differential?
Find the Gateaux differential for the functional $ S[y]=y(0)+\frac{1}{2}\int_{0}^{1}(4y'^2+x^2y^2)dx, y(1)=2 $ and use it to derive the Euler-Lagrange equation. Be sure to specify all boundary ...
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Derive Euler-Lagrange equation involving Dirac delta [closed]
I tried to derive Euler-Lagrange from a functional:
$E(\phi) = \int_{\Omega} |\nabla \phi | \delta(\phi) dx$
where $\phi$ is real-valued function depending on $x$. I denote $F(\phi, \nabla\phi) = |\...
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2
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0
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Functional derivative of line integral
If we can interpret the line integral of a function over a path as a functional. What would be its functional derivative?
For instance, in this example, let $\gamma$ be a path in $\mathbb{R}^3$. And ...
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0
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References for $\Gamma$-convergence
There are two books (that I know of) on $\Gamma$-convergence, including "An Introduction to $\Gamma$-convergence" by del Maso, and "A Handbook of $\Gamma$-convergence" by Andrea ...
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1
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Equivalent Formulations of Variational Problems
This post is supposed to collect some Theorems and techniques which can be used to analyse variational problems by (a) finding a related variational problem s.t. their optimal values are the same or ...
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1
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Deriving Lagrangian for common a class of PDEs
I am interested in constructing a Lagrangian for a PDE of type
$$
u_t(t,x) - F(u,u_t,u_x)=0
$$
such that for some functional $\quad I[u] = \int D[u]\mathcal{L}[u,u_t,u_x]$
its associated Euler ...
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0
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On the definition of pregeodesics as stationary points of the length functional in (pseudo-)Riemannian geometry
On a Riemannian manifold $(M,g)$, pregeodesics can be defined either of the following two equivalent ways:
curves for which the (Levi-Civita) acceleration $\nabla_{\dot\gamma(t)}\dot\gamma(t)$ is ...
- 1,653
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2
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Euler Lagrange equation of a functional on the space of traceless symmetric matrices
Let $\mathcal{S}_0:\{Q\in \mathbb{R}^{3\times 3}:\text{tr}Q=0 \hspace{5pt}\text{and}\hspace{5pt} Q_{ij}=Q_{ji} \hspace{5pt} \text{for any $i,j=1,2,3$}\}$ and
$$ \widetilde{ \mathcal{E}}(Q)
=\int_{\...
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0
answers
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Struggling to Derive First-Order Condition in Lucas (2004) on Optimal Control
I am reading Robert Lucas (2004), Life Earnings and Rural-Urban Migration, and I came across a rather peculiar optimal control problem that I'd like to ask about. Thank you!
The objective function is
$...
